Question

Usando o método de decomposição em frações parciais, integre a função:

∫ 3x+1/ x²-5x+6 dx

Answer 1

$\int\ {\dfrac{3x+1}{x^2-5x+6}} \, dx \\\\\\ \int\ {\dfrac{3x+1}{(x-2)\cdot(x-3)}} \, dx \\\\\\ \dfrac{3x+1}{(x-2)\cdot(x-3)}= \dfrac{A}{x-2}+ \dfrac{B}{x-3} \\\\\\ \dfrac{3x+1}{(x-2)\cdot(x-3)}= \dfrac{(x-3)\cdot{A}}{(x-2)}+\dfrac{(x-2)\cdot{B}}{(x-3)}\\\\\\3x+1=(x-3)\cdot{A}+(x-2)\cdot{B}\\3x+1=Ax-3A+Bx-2B\\3x+1=(A+B)x-3A-2B\\\\A+B=3\\-3A-2B=1\\\\\\\left \{ {{A+B=3} \atop {-3A-2B=1}} \right. \\\\3A+3B=3\\-3A-2B=1\\\\B=4\\\\A+B=3\\A+4=3\\A=-1$

$\dfrac{3x+1}{(x-2)\cdot{(x-3)}}= \dfrac{-1}{x-2}+\dfrac{4}{x-3}\\\\\\ \int\ {\bigg(\dfrac{-1}{x-2}+\dfrac{4}{x-3}}\bigg)\,dx\\\\\\ =\int{\dfrac{-1}{x-2}}\, dx +\int{\dfrac{4}{x-3}}\,dx\\\\\\=-ln(x-2)+4ln(x-3)+C$